Hadwiger conjectured in 1943 that every graph with no $K_t$ minor has chromatic number at most $t-1$. Delcourt and Postle proved that every graph with no $K_t$ minor has chromatic number $O(t\log\log t)$. We build on their result to improve this bound to $O(t\log\log\log t)$.
We show that there exists $C\in\mathbb{N}$ such that $K_t$-minor free graphs are $Ct$-colorable. The proof was found by GPT-6 Astra, following the directions by the authors.
Delcourt and Postle proved that every $K_t$-minor-free graph is $O(t\log\log t)$-colorable and reduced the Linear Hadwiger Conjecture to coloring $K_t$-minor-free graphs on $O(t\log^4 t)$ vertices. In this paper, we use the star contraction technique to improve their bound for small graphs and use their reduction to ex...
A graph $G$ with 243 vertices is shown having $\gamma(G) = \gamma^{\infty}(G)<\theta(G)$, disproving the Gamma-Theta conjecture for eternal dominating sets.
The linear list Hadwiger conjecture asserts that every $K_t$-minor-free graph is $O(t)$-choosable. Following an approach outlined by Delcourt and Postle, we recently established $O(t\log\log t)$-choosability. Soon afterward, OpenAI provided a proof of the conjecture. A key new ingredient of their proof is a linear boun...
Erd\H{o}s and S\'os conjectured that every $n$-vertex $3$-uniform hypergraph whose link graphs are all bipartite has at most $(1/4+o(1))\binom n3$ edges. We disprove this conjecture by constructing counterexamples with edge density at least $0.250000356>1/4$ for all sufficiently large $n$.
For every $n$, we construct a 2-dimensional $n$-element poset whose Hasse diagram has independence number $n\exp(-\Omega(\sqrt{\log n}))$, and consequently, chromatic number $\exp(\Omega(\sqrt{\log n}))$. This also yields an $n$-point planar set whose box-Delaunay graph satisfies the same bounds.
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